Laboratoire de l'Informatique et du Parallélisme » Pobyso

def slz_compute_polynomial_and_interval(functionSo, degreeSo, lowerBoundSa,

                                        upperBoundSa, approxPrecSa,

                                        sollyaPrecSa=None):

    """

    Under the assumptions listed for slz_get_intervals_and_polynomials, compute

    a polynomial that approximates the function on a an interval starting

    at lowerBoundSa and finishing at a value that guarantees that the polynomial

    approximates with the expected precision.

    The interval upper bound is lowered until the expected approximation

    precision is reached.

    The polynomial, the bounds, the center of the interval and the error

    are returned.

    """

    RRR = lowerBoundSa.parent()

    #goldenRatioSa = RRR(5.sqrt() / 2 - 1/2)

    #intervalShrinkConstFactorSa = goldenRatioSa

    intervalShrinkConstFactorSa = RRR('0.5')

    absoluteErrorTypeSo = pobyso_absolute_so_so()

    currentRangeSo = pobyso_bounds_to_range_sa_so(lowerBoundSa, upperBoundSa)

    currentUpperBoundSa = upperBoundSa

    currentLowerBoundSa = lowerBoundSa

    # What we want here is the polynomial without the variable change,

    # since our actual variable will be x-intervalCenter defined over the

    # domain [lowerBound-intervalCenter , upperBound-intervalCenter].

    (polySo, intervalCenterSo, maxErrorSo) = \

        pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo,

                                                    currentRangeSo,

                                                    absoluteErrorTypeSo)

    maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

    while maxErrorSa > approxPrecSa:

        sollya_lib_clear_obj(maxErrorSo)

        errorRatioSa = (approxPrecSa/maxErrorSa).log2()

        #print "Error ratio: ", errorRatioSa

        if errorRatioSa < intervalShrinkConstFactorSa:

            #currentUpperBoundSa = currentLowerBoundSa + (currentUpperBoundSa - currentLowerBoundSa) * errorRatioSa

            currentUpperBoundSa = currentLowerBoundSa + \

                                  (currentUpperBoundSa - currentLowerBoundSa) * \

                                   intervalShrinkConstFactorSa

        else:

            currentUpperBoundSa = currentLowerBoundSa + \

                                  (currentUpperBoundSa - currentLowerBoundSa) * \

                                  intervalShrinkConstFactorSa

        #print lowerBoundSa, currentUpperBoundSa

        sollya_lib_clear_obj(currentRangeSo)

        sollya_lib_clear_obj(polySo)

        currentRangeSo = pobyso_bounds_to_range_sa_so(currentLowerBoundSa,

                                                      currentUpperBoundSa)

        (polySo, intervalCenterSo, maxErrorSo) = \

            pobyso_taylor_expansion_no_change_var_so_so(functionSo, degreeSo,

                                                        currentRangeSo,

                                                        absoluteErrorTypeSo)

        #maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo, RRR)

        maxErrorSa = pobyso_get_constant_as_rn_with_rf_so_sa(maxErrorSo)

    sollya_lib_clear_obj(absoluteErrorTypeSo)

    return((polySo, currentRangeSo, intervalCenterSo, maxErrorSo))

    # End slz_compute_polynomial_and_interval

def slz_get_intervals_and_polynomials(functionSa, variableNameSa, degreeSa,

                                      lowerBoundSa,

                                      upperBoundSa, floatingPointPrecSa,

                                      internalSollyaPrecSa, approxPrecSa):

    """

    Under the assumption that:

    - functionSa is monotonic on the [lowerBoundSa, upperBoundSa] interval;

    - lowerBound and upperBound belong to the same binade.

    from a:

    - function;

    - a degree

    - a pair of bounds;

    - the floating-point precision we work on;

    - the internal Sollya precision;

    - the requested approximation error

    compute a list of tuples made of:

    - a polynomial approximating the function (a Sollya object);

    - the range for which the polynomial approximates the function

      (a Sollya object);

    - the center of the interval (a Sollya object);

    - the actual approximation error (a Sage object).

    The initial interval is, possibly, splitted into smaller intervals.

    It return a list of tuples, each made of:

    - a polynomial;

    - the approximation interval;

    - the center, x0,  of the interval (the polynomial is defined as p(x-x0));

    - the corresponding approximation error.

    """

    x = var(variableNameSa)

    # Scalling the domain -> [1,2[.

    boundsIntervalRifSa = RealIntervalField(floatingPointPrecSa)

    domainBoundsIntervalSa = boundsIntervalRifSa(lowerBoundSa, upperBoundSa)

    (domainScalingExpressionSa, invDomainScalingExpressionSa) = \

        slz_interval_scaling_expression(domainBoundsIntervalSa, variableNameSa)

    print "domainScalingExpression for argument :", domainScalingExpressionSa

    print "f: ", f

    ff = f.subs({x : domainScalingExpressionSa})

    #ff = f.subs_expr(x==domainScalingExpressionSa)

    scaledLowerBoundSa = invDomainScalingExpressionSa(lowerBoundSa).n()

    scaledUpperBoundSa = invDomainScalingExpressionSa(upperBoundSa).n()

    print 'ff:', ff, "- Domain:", scaledLowerBoundSa, scaledUpperBoundSa

    #

    # Scalling the image -> [1,2[.

    flbSa = f(lowerBoundSa).n()

    fubSa = f(upperBoundSa).n()

    if flbSa <= fubSa: # Increasing

        imageBinadeBottomSa = floor(flbSa.log2())

    else: # Decreasing

        imageBinadeBottomSa = floor(fubSa.log2())

    print 'ff:', ff, '- Image:', flbSa, fubSa, imageBinadeBottomSa

    imageBoundsIntervalSa = boundsIntervalRifSa(flbSa, fubSa)

    (imageScalingExpressionSa, invImageScalingExpressionSa) = \

        slz_interval_scaling_expression(imageBoundsIntervalSa, variableNameSa)

    iis = invImageScalingExpressionSa.function(x)

    fff = iis.subs({x:ff})

    print "fff:", fff,

    print " - Image:", fff(scaledLowerBoundSa), fff(scaledUpperBoundSa)

    #

    resultArray = []

    #

    print "Approximation precision: ", RR(approxPrecSa)

    # Prepare the arguments for the Taylor expansion computation with Sollya.

    functionSo = pobyso_parse_string_sa_so(fff._assume_str())

    degreeSo = pobyso_constant_from_int_sa_so(degreeSa)

    scaledBoundsSo = pobyso_bounds_to_range_sa_so(scaledLowerBoundSa,

                                                  scaledUpperBoundSa)

    # Compute the first Taylor expansion.

    (polySo, boundsSo, intervalCenterSo, maxErrorSo) = \

        slz_compute_polynomial_and_interval(functionSo, degreeSo,

                                        scaledLowerBoundSa, scaledUpperBoundSa,

                                        approxPrecSa, internalSollyaPrecSa)

    # Change variable stuff

    changeVarExpressionSo = sollya_lib_build_function_sub(

                              sollya_lib_build_function_free_variable(),

                              sollya_lib_copy_obj(intervalCenterSo))

    polyVarChangedSo = sollya_lib_evaluate(polySo, changeVarExpressionSo)

    resultArray.append((polySo, polyVarChangedSo, boundsSo, intervalCenterSo,\

                         maxErrorSo))

    realIntervalField = RealIntervalField(max(lowerBoundSa.parent().precision(),

                                              upperBoundSa.parent().precision()))

    boundsSa = pobyso_range_to_interval_so_sa(boundsSo, realIntervalField)

    # Compute the other expansions.

    while boundsSa.endpoints()[1] < scaledUpperBoundSa:

        currentScaledLowerBoundSa = boundsSa.endpoints()[1]

        (polySo, boundsSo, intervalCenterSo, maxErrorSo) = \

            slz_compute_polynomial_and_interval(functionSo, degreeSo,

                                            currentScaledLowerBoundSa,

                                            scaledUpperBoundSa, approxPrecSa,

                                            internalSollyaPrecSa)

        # Change variable stuff

        changeVarExpressionSo = sollya_lib_build_function_sub(

                                  sollya_lib_build_function_free_variable(),

                                  sollya_lib_copy_obj(intervalCenterSo))

        polyVarChangedSo = sollya_lib_evaluate(polySo, changeVarExpressionSo)

        resultArray.append((polySo, polyVarChangedSo, boundsSo, \

                            intervalCenterSo, maxErrorSo))

        boundsSa = pobyso_range_to_interval_so_sa(boundsSo, realIntervalField)

    sollya_lib_clear_obj(functionSo)

    sollya_lib_clear_obj(degreeSo)

    sollya_lib_clear_obj(scaledBoundsSo)

    return(resultArray)

    # End slz_get_intervals_and_polynomials

def slz_interval_scaling_expression(boundsInterval, varName):

    """

    Compute the scaling expression to map an interval that span only

    a binade to [1, 2) and the inverse expression as well.

    Not very sure that the transformation makes sense for negative numbers.

    """

    # The scaling offset is only used for negative numbers.

    if abs(boundsInterval.endpoints()[0]) < 1:

        if boundsInterval.endpoints()[0] >= 0:

            scalingCoeff = 2^floor(boundsInterval.endpoints()[0].log2())

            invScalingCoeff = 1/scalingCoeff

            return((scalingCoeff * eval(varName),

                    invScalingCoeff * eval(varName)))

        else:

            scalingCoeff = \

                2^(floor((-boundsInterval.endpoints()[0]).log2()) - 1)

            scalingOffset = -3 * scalingCoeff

            return((scalingCoeff * eval(varName) + scalingOffset,

                    1/scalingCoeff * eval(varName) + 3))

    else:

        if boundsInterval.endpoints()[0] >= 0:

            scalingCoeff = 2^floor(boundsInterval.endpoints()[0].log2())

            scalingOffset = 0

            return((scalingCoeff * eval(varName),

                    1/scalingCoeff * eval(varName)))

        else:

            scalingCoeff = \

                2^(floor((-boundsInterval.endpoints()[1]).log2()))

            scalingOffset =  -3 * scalingCoeff

            #scalingOffset = 0

            return((scalingCoeff * eval(varName) + scalingOffset,

                    1/scalingCoeff * eval(varName) + 3))

def slz_polynomial_and_interval_to_sage(polyRangeCenterErrorSo):

    polynomialSa = pobyso_get_poly_so_sa(polyRangeCenterErrorSo[0])

    polynomialChangedVarSa = pobyso_get_poly_so_sa(polyRangeCenterErrorSo[1])

    intervalSa = \

            pobyso_get_interval_from_range_so_sa(polyRangeCenterErrorSo[2])

    centerSa = \

            pobyso_get_constant_as_rn_with_rf_so_sa(polyRangeCenterErrorSo[3])

    errorSa = \

            pobyso_get_constant_as_rn_with_rf_so_sa(polyRangeCenterErrorSo[4])

    return((polynomialSa, polynomialChangedVarSa, intervalSa, centerSa, errorSa))

    # End slz_polynomial_and_interval_to_sage

print "sageSLZ loaded..."